Ein Wohlordnungsbeweis für das OrdinalzahlensystemT(J)

Summary A recursive notation system of a strong segment of ordinals was developped by Jäger [3]. An unessential modified versionT(J) of this notation system was described in [4]. In the following, the well-ordering ofT(J) is proved in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension. It follows, that the proof theoretical ordinal of ₂ ¹ -analysis is greater than the order type ofT(J).     

Eine ErweiterungT(V′) des Ordinalzahlensystems 58-0158-0158-01(Λ0) von G. Jäger

Summary This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V) is provable in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V).     

Controlled blocks of the finite quasisimple groups for odd primes

We classify controlled blocks, introduced by Alperin and Broué in 1979 for all quasisimple groups G for odd primes. The results imply that every nilpotent block of G has abelian defect groups, which in turn is one of the main results proved in An and Eaton (2011) [6]. We also give an explicit characterization of non-controlled blocks of all quasisimple groups G for odd primes. This implies the block theoretic analogue of Glauberman?s ZJ-theorem for G proved by Kessar, Linckelmann and Robinson (2002) [18].     

Extended finite operator calculus—an example of algebraization of analysis

“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota—Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form—a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota—Mullin or equivalently—of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]).     

Testing significance of a mean vector—A possible alternative to Hotelling'sT 2

In problems involving multivariate measurements experimental considerations often indicate grouping of variables into subsets ordered according to their importance. In such situations, the problems such as comparison of two mean vectors and profile analysis may be treated by Hotelling'sT ²-test adapted along the lines of the step-wise procedure of J. Roy [10], or the well known test for additional information due to Rao [9]. In this paper we study a modification of the step-wise procedure obtained by combining the component tests. The exact Bahadur slopes of resulting procedures are computed and it is shown that the procedure based upon Fisher's combination method is asymptotically equivalent to Hotelling'sT ². A Monte Carlo study suggests that even in small samples the power functions of the new method and Hotelling'sT ²-test are practically equivalent. Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant No. AFOSR-77-3360. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.     

Hodge genera of algebraic varieties I

The aim of this paper is to study the behavior of Hodge‐theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13, 32]. © 2007 Wiley Periodicals, Inc.     

Schensted Algorithms for Dual Graded Graphs

This paper is a sequel to [3]. We keep the notation and terminology and extend the numbering of sections, propositions, and formulae of [3].The main result of this paper is a generalization of the Robinson-Schensted correspondence to the class of dual graded graphs introduced in [3], This class extends the class of Y-graphs, or differential posets [22], for which a generalized Schensted correspondence was constructed earlier in [2].The main construction leads to unified bijective proofs of various identities related to path counting, including those obtained in [3]. It is also applied to permutation enumeration, including rook placements on Ferrers boards and enumeration of involutions.As particular cases of the general construction, we re-derive the classical algorithm of Robinson, Schensted, and Knuth [19, 12], the Sagan-Stanley [18], Sagan-Worley [16, 29] and Haiman's [11] algorithms and the author's algorithm for the Young-Fibonacci graph [2]. Some new applications are suggested.The rim hook correspondence of Stanton and White [23] and Viennot's bijection [28] are also special cases of the general construction of this paper.In [5], the results of this paper and the previous paper [3] were presented in a form of extended abstract.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Corrigendum to the paper “Hermitian forms over division algebras over real function fields”

In the previous paper [T] we gave a classification of hermitian forms over the real function fieldk=R(t) and its completionsk _v with respect to valuationsv trivial onR. Unfortunately in the local case the arguments given for cases A and D, in general, were not correct. Therefore the resulting local and local-global classifications obtained were incorrect. I would like also to thank Dr. D. Hoffmann for pointing out these mistakes and the referee for useful comments. Here we would like to make necessary corrections to [T]. We keep the same notation used there, except that in the first paragraph,J is not the standard involution of a quaternion division algebraD (with basis {1,i,j,ij}). All hermitian forms will be hermitian forms with respect toJ, with values inD.     

Geometric versus homotopy theoretic equivariant bordism

By results of Löffler and Comezaña, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S¹××S¹, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.     

Cluster Categories and Their Relation to Cluster Algebras and Semi-invariants

Cluster categories were introduced in the paper “Tilting theory and cluster combinatorics” [8] in order to better understand the combinatorics of cluster algebras, by giving new, module theoretic and categorical meanings to the combinatorics of the well known cluster algebras [20]. Subsequently, correspondences between the many notions in these two areas were given, e.g. [5, 6, 11, 9] and others. This proved to be quite useful and productive approach with even further connections to semi-invariants of quivers [26]. However, in order to get this connection, one needs to define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. Then, the virtual semi-invariants satisfy the three basic theorems: the First Fundamental Theorem (determinantal), the Saturation theorem and the Canonical Decomposition theorem. From the above theorems it follows that in the case of Dynkin quivers there is a nice relationship between supports of the semi-invariants and the tilting triangulation of the (n – 1)-sphere. Lecture held in the Seminario Matematico e Fisico on September 26, 2007. Received: September 2008     

Dynamic ordinal analysis

 Dynamic ordinal analysis is ordinal analysis for weak arithmetics like fragments of bounded arithmetic. In this paper we will define dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠ ^b ₁(X) order induction available in a theory. We will compare order induction to successor induction over weak theories. We will compute dynamic ordinals of the bounded arithmetic theories sΣ ^b _n (X)−L ^m IND for m=n and m=n+1, n≥0. Different dynamic ordinals lead to separation. In this way we will obtain several separation results between these relativized theories. We will generalize our results to further languages extending the language of bounded arithmetic. Received: 27 April 2001 / Published online: 19 December 2002 The results for sΣ ^b _n (X)−L ^m IND are part of the authors dissertation [3]; the results for sΣ ^b _m (X)−L ^m+1 IND base on results of ARAI [1]. Mathematics Subject Classification (2000): Primary 03F30; Secondary 03F05, 03F50 Key words or phrases: Dynamic ordinal – Bounded arithmetic – Proof-theoretic ordinal – Order induction – Semi-formal system – Cut-elimination     

Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method

Abstract The pointwise gradient constrained homogenization process, for Neumann and Dirichlet type problems, is analyzed by means of the periodic unfolding method recently introduced in [21]. Classically, the proof of the homogenization formula in presence of pointwise gradient constraints relies on elaborated measure theoretic arguments. The one proposed here is elementary: it is based on weak convergence arguments in L^p spaces, coupled with suitable regularization techniques. Keywords: Homogenization, Gradient constrained problems, Periodic unfolding method Mathematics Subject Classification (2000): 49J45, 35B27, 74Q05     

Inconsistency of GPK + AFA

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X₁ introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X₁ in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. Mathematics Subject Classification: 03E70, 03E10.     

Some estimates of the normal approximation for independent non-identically distributed random variables

In this paper, we generalize some results of [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003] for independent identically distributed summands to to the case of independent non-identically distributed real summands. We derive the Edgeworth expansion with the first term only. Proofs are given following [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003].     

On the existence theorem for the barotronic motion of a compressible inviscid fluid in the half-space

Summary We give here an existence theorem, see Theorem 1.1,for the solution of the system of equations (1.1)that describes the motion of a compressible inviscid fluid in the half space. Moreover, we establish some sharp estimates, see Theorem 3.2,for the solution of the linear second order hyperbolic mixed problem (3.1)in terms of suitable norms of the coefficients. These estimates play a main rule here and in reference [BV3],where a first proof of Hadamard's classical well-posedness for the above nonlinear system of equations is given; see also [BV4].Here, we adapt and simplify the method followed in our previous paper [BV1].     

Algebras over variable theories

We will show that algebras over variable theories form a variety. This variety interprets the type Self of object oriented programming proposed by Abadi and Cardelli in [1]. Received February 14, 2000; accepted in final form April 5, 2001.     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.